Enhanced Power Extraction via Hybrid Pitching Motion in an Oscillating Wing Energy Harvester with Leading Flap

Abstract

This study applied a hybrid pitching motion for an oscillating wing with a leading flap aimed at enhancing energy extraction efficiency. In the first half of the cycle, the hybrid pitching motion begins with a non-sinusoidal pitching motion for 0.0 ≤ t/T ≤ 0.25, transitioning to a sinusoidal pitching motion for 0.25 < t/T ≤ 0.50. The latter half of the motion mirrors the first one but moves toward the reverse direction. Hybrid motions combine the benefits of non-sinusoidal and sinusoidal pitching motions, enhancing the optimization of pitch angle variation. The findings show that hybrid motions for the wing fitted with an attached leading flap outperform both the single plate and the wing with an attached flap using sinusoidal pitching motion. The simulation was conducted with flap lengths ranging from 30% to 45% of the chord length and examined maximum pitching angles of the wing and the attached leading flap between 80° to 95° and 25° to 60°, respectively. By setting the pitch angles of the wing and leading flap to 85° and 45°, respectively, with the wing comprising 65% of the total length and the leading flap 35%, the proposed hybrid pitching motion with the leading flap generates a maximum power output of 1.276 that surpasses that of a sinusoidal pitching motion of 0.963 on an oscillating flat plate by 32.50%. This combination of hybrid pitching motion and a wing flap configuration is effective in improving the performance.

1. Introduction

Oscillating foils provide an eco-friendly alternative to conventional rotary turbines for extracting energy from fluid flow, contributing to sustainable power generation and minimizing environmental impact. Transitioning from conventional fossil fuels to renewable energy sources is pushing forward new breakthroughs within the energy field. One notable development includes systems using oscillating foils, known for their eco-friendly and efficient conversion of kinetic energy from fluid currents, converting it to practical energy [1,2]. While the study of energy capture through oscillating fluid-dynamic systems is relatively new, the concept of using flapping motions for energy production was initially proposed in 1972 [3], and its foundational principles were outlined in 1981 [4]. Their research concentrated on harmonically oscillating wings, using a combination of pitching and plunging motions to capture wind energy, and achieved efficiencies similar to conventional wind turbines. Extensive research has built upon initial studies to improve efficiency for flapping foils. Davids [5] conducted a study on a NACA0012 airfoil, demonstrating that optimal frequency and plunging amplitude can achieve an efficiency of 30%. Research on a NACA0015 revealed that tuning the pitching motion to specific frequency and amplitude ranges resulted in 34% efficiency, demonstrating that this approach outperforms simple plunging in enhancing performance [6]. Another investigation revealed that the effectiveness of flapping foil is influenced by various parameters, including the dynamics of foil movement, the configuration of the system, the mechanics of the flow, and the trajectory of the motion [7,8].

He et al. [9] observed that, in tandem configurations, the upstream foil consistently captures energy irrespective of the spacing between foils. However, as the separation distance increases, the downstream foil undergoes cyclical fluctuations in performance. Numerical analysis has demonstrated that a new tandem wing design flapping within a tapering duct greatly increases power [10]. Dahmani and Sohn [11] found that adjusting the vertical distance between oscillating wings in tandem or parallel configurations can significantly enhance energy capture, resulting in a 23% increase in efficiency. Furthermore, a concept for a tandem hydrodynamic foil tidal system has been proposed, highlighting benefits in maximizing capacity density and reducing electricity costs [12]. Investigations into optimal propulsion patterns for small aerial drones and autonomous underwater vehicles have emphasized the link between variations in the angle of attack and propulsion performance [13]. Studies on oscillating wings have delved into numerous deflector configurations aimed at boosting power. Investigations underline the critical roles of optimizing attack angle and strategically positioning deflectors relative to the incoming flow to enhance the efficiency of power generation [14].

Passive control methods have been shown to enhance efficiency by applying adjustments to an airfoil’s surface, thereby removing the need for any additional energy supply. Yin and Luo [15] suggest that flexible bending on the foil’s suction side enhances vortex dynamics, leading to increased thrust force and improved energy output. A fully passive airborne wind energy system successfully harvested wind energy without relying on actuators or motors, improving the performance [16]. The integration of a Gurney flap has been identified as a strategy to enhance energy harvesting efficiency by promoting increased vortex formation near the rear edge and amplifying pressure differentials. This enhancement significantly boosts heaving force and yields a 21% boost in efficiency [17,18]. Sun et al. [19] investigated the use of movable Gurney flaps to optimize energy harvesting under diverse flow conditions. Adding a cylindrical spoiler at the foil’s trailing edge boosted efficiency by 19.26% higher than configurations without any spoiler [20]. Petikidis and Papadakis. [21] investigated the behavior of fully passive oscillating foils, examining the optimal submersion depths for maximum performance. Recent developments in control systems have focused on optimizing aerodynamic performance by adjusting the attack angle aimed to enhance lift force and improve overall power output [22]. Research has shown that incorporating shrouds can boost efficiency by 35.8% relative to a design without a shroud [23]. Attaching flaps to the wings has been found to affect energy extraction efficiency in rigid and flexible plates [24].

Active control methods have been found to be more effective than passive methods [25,26]. Flaps are commonly employed in the aerospace sector for lifting mechanisms because of their simple design, robustness, and effectiveness [27,28]. Implementing a completely flexible flapping wing resulted in a 2% improvement in power output compared to a conventional airfoil design [29]. Alam and Sohn [30] reported an 11% increase in power output for an oscillating wing with a trailing flap compared to a wing without a flap. They [31] found that extending the trailing flap of a flapping wing resulted in a 26.9% increase in power and a 21% improvement in efficiency. They also conducted further studies on integrating leading flaps into oscillating wing harvesters, identifying a configuration that increased power output by 29.9% and improved efficiency by 23% [32]. The investigation revealed that the implementation of trailing-edge jet flaps improves the energy harvesting efficiency of flapping foils [33].

Observational findings suggest that targeted structural modifications can greatly improve the performance of energy-harvesting systems [34]. Maruai et al. [35] investigated the impact of modifying flow dynamics to enhance vibrational energy output. Studies have investigated the impact of dynamic wing configuration adjustments on energy extraction by altering lift and drag forces [36,37]. Saleh and Sohn [38] investigated an oscillating wing with both leading and trailing flaps using sinusoidal motion. Through examining various configurations, their study achieved a 28.24% increase in power output compared to wings without flaps. Mo et al. [39], Yang et al. [40], and Zhu et al. [41] investigated the intricate link between ground proximity and power output in flapping foils. As the distance between the foil and the ground reduces, power production rises due to increased plunging motion and lift; however, too close proximity might eventually degrade performance. Yang et al. [42] further examined how single-sided and double-sided wall confinement affects flapping foils, finding that a single wall improves energy harvesting compared to an unconfined setting.

Swain et al. [43] demonstrated that combining pitching and heaving motions can significantly enhance performance. By adjusting foil profiles and placements, they have focused on optimizing wake interactions to maximize overall efficiency. The performance of flapping hydrofoils was measured through various types of motion. The motion of left-swing outperformed the other movements in terms of power extraction [44]. Studies indicate that non-sinusoidal motions, particularly in pitching and plunging movements, significantly enhance energy extraction from flapping foils, demonstrating maximum power output when compared to sinusoidal counterparts [45,46]. Deng et al. [47] indicated that sinusoidal motions can achieve efficiencies greater than 33% for flapping foils at a pitch angle of 75°. These findings highlight the effectiveness of sinusoidal pitching motion in improving energy efficiency in flapping foil systems. Teng et al. [48] explored the impact of non-sinusoidal motion on efficiency, finding that a pitching angle of 75° achieved an optimal efficiency of 32%. Li et al. [49] investigated the effectiveness of semi-active foils using cosine motions to enhance power efficiency. Their findings demonstrated that these foils significantly boost energy harvesting efficiency. Lv and Sun [50] explored the potential benefits of incorporating a modified trailing edge airfoil with a jet flap in a semi-active flapping foil, achieving a 1.46% increase in energy harvesting efficiency compared to traditional designs. Recent research has explored hybrid motion that integrates a non-sinusoidal pitching motion between time 0.0 and 0.25, followed by a sinusoidal pitching motion from 0.25 to 0.5. It was found that this hybrid approach yielded a total power coefficient of 1.16 [51]. Zheng et al. [52] examined the power extraction capabilities of a flapping-foil turbine following a figure-eight trajectory, finding that this motion significantly enhances energy harvesting efficiency. Tian and Liu [53] revealed that incorporating movable lateral flaps on the pressure side (PMLFs) led to a 21.7% improvement in energy extraction performance compared to conventional foils.

Previous studies focused on sinusoidal and non-sinusoidal motion on a single wing and on a wing with one or two flaps. Their studies indicated that merging one or two flaps notably boosts performance. Our study examines the effectiveness of hybrid pitching motion configurations to enhance performance, including applying hybrid pitching motion to both the wing and leading flap, applying hybrid motion to the wing with sinusoidal motion on the leading flap, and applying sinusoidal motion to the wing with hybrid pitching on the leading flap. A computational study was conducted to evaluate the impact of different pitch angles and lengths of a wing with an attached leading flap on overall performance. Pitch angles of 80°, 85°, 90°, and 95° were selected for the wing, as these angles were observed to produce higher performance.

2. Numerical Methodology

A flat plate offers advantages that exceed those of a traditional profiled airfoil [54]. Figure 1 shows a two-dimensional oscillating wing with an attached leading flap. The total chord length is c = 1.0 with a thickness (${\mathrm{t}}_{\mathrm{w}}$) equal to 4% of c, a heaving amplitude of ${\mathrm{h}}_{\mathrm{o}}$/c = 1.0, and a reduced frequency of $f^{\ast }$ = 0.14; optimal power and performance are achieved [54]. The lengths and pitch angles for the wing with an attached leading flap have been adjusted. The gap between the wing and the attached flap is set at 0.002, and the radius of the flat plate is 0.5 times the thickness of the wing. Figure 2 presents a simplified schematic of the flapping wing equipped with a leading flap. The following symbolic variables are introduced: $x_p$, the pivot point, located at one-third of the chord length c from the leading edge; ${\mathrm{U}}_{\mathrm{\infty }}$, the velocity of the incoming flow; h(t), the instantaneous vertical displacement of the flat plate; ${\mathrm{h}}_0$, the heaving amplitude; ω = 2πf, the angular frequency; f, the oscillation frequency; ∅, the phase angle; and t, the time.

$$\mathrm{h}\left(\mathrm{t}\right)={\mathrm{h}}_0 \sin (\omega \mathrm{t} + \varnothing )$$

(1)

Figure 3 illustrates the variation in the instantaneous pitching angle over one cycle. The sinusoidal motion is represented by a black line, as shown in Figure 3a. Figure 3b,c depict the non-sinusoidal pitching motion and hybrid pitching motion, respectively, both presented as functions of the parameter β. The sinusoidal motion is defined in Equation (2), the non-sinusoidal motion in Equation (3), and the hybrid motion in Equation (4).

In this study, a value of β = 1.5 was selected for the hybrid pitching motion, as it provides optimal performance at this specific setting [51], with a phase lag of $\varnothing$ = 90° relative to the heaving motion. Three cases were explored: the first case examines the effects of applying hybrid pitching motion to both the wing and leading flap, as described in Equation (4). The second case involves using sinusoidal motion to the leading flap while applying hybrid motion for the wing, as defined in Equations (2) and (4). In the third case, wing undergoes sinusoidal motion, while hybrid motion is applied to the leading flap, as outlined in Equations (2) and (4).

Let $\mathrm{S}$(t) represent the instantaneous pitching angle, where for the wing, $\mathrm{S}$(t) = $\theta$(t) and ${\mathrm{S}}_{\mathrm{o}}$ = ${\theta }_{\mathrm{o}}$, and for the leading flap, $\mathrm{S}$(t) = $\psi$(t) and ${\mathrm{S}}_{\mathrm{o}}$ = ${\psi }_{\mathrm{o}}$.

$$\mathrm{S}\left(\mathrm{t}\right)={\mathrm{S}}_{\mathrm{o}} sin (\omega t)$$

(2)

$$\mathrm{S}\left(\mathrm{t}\right)={\mathrm{S}}_{\mathrm{o}} tanh [\beta sin (\omega t\left)\right]/tanh(\beta )$$

(3)

In hybrid profiles, the foil first undergoes a rapid ascent to its peak pitch angle following a non-sinusoidal profile from t = 0.0 to t = 0.25. After this initial phase, the motion transitions to a sinusoidal pattern, continuing from the midpoint of the ascent until the foil returns to a horizontal orientation between t = 0.25 and t = 0.50.

$$\begin{array}{l}\mathrm{S}\left(\mathrm{t}\right)=\left\{\begin{array}{l}{\mathrm{S}}_{\mathrm{o}} \tanh [\beta \sin (\omega \mathrm{t}\left)\right]/\tanh (\beta ) 0.0\le \mathrm{t}/\mathrm{T}\le 0.25\\ {\mathrm{S}}_{\mathrm{o}} \sin (\omega \mathrm{t}) 0.25\le \mathrm{t}/\mathrm{T}\le 0.5\\ {\mathrm{S}}_{\mathrm{o}} \tanh [\beta \sin (\omega \mathrm{t}\left)\right]/\tanh (\beta ) 0.5\le \mathrm{t}/\mathrm{T}\le 0.75\\ {\mathrm{S}}_{\mathrm{o}} \sin (\omega \mathrm{t}) 0.75\le \mathrm{t}/\mathrm{T}\le 1.0\end{array}\right.\\ \end{array}$$

(4)

$$\mathrm{Re}={\displaystyle \frac{\rho {\mathrm{U}}_{\mathrm{\infty }}\mathrm{c}}{\mathsf{\mu }}}$$

(5)

where $\rho$ represents the density, ${\mathrm{U}}_{\mathrm{\infty }}$ is the uniform speed, μ denotes the dynamic viscosity, and f* is the reduced frequency. Two-dimensional unsteady simulations were conducted at Reynolds number ${\mathrm{R}}_{\mathrm{e}}$ = 500,000.

$$f^{\ast }={\displaystyle \frac{f\mathrm{c}}{{\mathrm{U}}_{\mathrm{\infty }}}}$$

(6)

The computation methods for average power ($\overline{\mathrm{P}}$) and instantaneous power (P) are detailed as follows:

$$\mathrm{P}=\mathrm{Y}\left(\mathrm{t}\right)\left[\mathrm{d}\mathrm{h}\right(\mathrm{t})/\mathrm{d}\mathrm{t}]+\mathrm{M}\left(\mathrm{t}\right) [\mathrm{d}\theta (\mathrm{t})/\mathrm{d}\mathrm{t}]$$

(7)

$$\overline{\mathrm{P}}=1/\mathrm{T}{\int }_0^{\mathrm{T}}\mathrm{P} \mathrm{d}\mathrm{t} ,$$

(8)

where Y(t) denotes the force in the vertical direction and M(t) denotes the pitching moment.

$${\mathrm{C}}_{\mathrm{p}\mathrm{t}}={\displaystyle \frac{\mathrm{P}}{\frac{1}{2}\rho {\mathrm{U}}_{\mathrm{\infty }}^3\mathrm{s}\mathrm{c}}}={\displaystyle \frac{2}{\rho {\mathrm{U}}_{\mathrm{\infty }}^3\mathrm{s}\mathrm{c}}}[\mathrm{Y}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathrm{h}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}+\mathrm{M}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\theta \left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}]$$

(9)

$${\mathrm{C}}_{\mathrm{p}\mathrm{t}}={\mathrm{C}}_{\mathrm{p}\mathrm{l}}+{\mathrm{C}}_{\mathrm{p}\mathrm{m}}={\displaystyle \frac{1}{{\mathrm{U}}_{\mathrm{\infty }}}}[{\mathrm{C}}_{\mathrm{L}}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathrm{h}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}+{\mathrm{C}}_{\mathrm{M}}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\theta \left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}]$$

(10)

where s is the span of the flat plate, and the computation of the instantaneous coefficients ${\mathrm{C}}_{\mathrm{L}}$(t) and ${\mathrm{C}}_{\mathrm{M}}$(t) is detailed as follows:

$${\mathrm{C}}_{\mathrm{L}}(\mathrm{t})=\mathrm{Y}(\mathrm{t})/{\displaystyle \frac{1}{2}}\rho {\mathrm{U}}_{\mathrm{\infty }}^2\mathrm{c} \mathrm{s}$$

(11)

$${\mathrm{C}}_{\mathrm{M}}(\mathrm{t})=\mathrm{M}(\mathrm{t})/{\displaystyle \frac{1}{2}}\rho {\mathrm{U}}_{\mathrm{\infty }}^2\mathrm{c} \mathrm{s} \mathrm{s}$$

(12)

The computation of the average power coefficient $(\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{t}}})$ is obtained by integrating ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$.

$$\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{t}}}={\displaystyle \frac{1}{\mathrm{T}}}\underset{0}{\overset{\mathrm{T}}{\int }}{\mathrm{C}}_{\mathrm{p}\mathrm{t}}\mathrm{d}\mathrm{t}=\overline{\mathrm{P}}/({\displaystyle \frac{1}{2}}\rho {\mathrm{U}}_{\mathrm{\infty }}^3\mathrm{c})$$

(13)

or

$$\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{t}}}=\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{l}}}+\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{m}}}={\displaystyle \frac{1}{\mathrm{T}{\mathrm{U}}_{\mathrm{\infty }}}}[{\int }_0^{\mathrm{T}}{\mathrm{C}}_{\mathrm{L}}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathrm{h}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}\mathrm{d}\mathrm{t}+{\int }_0^{\mathrm{T}}{\mathrm{C}}_{\mathrm{M}}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\theta \left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}\mathrm{d}\mathrm{t}]$$

(14)

Equations (10) and (14) show that ${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$ and $\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{l}}}$ correspond to the energy extraction from pushing forces, while ${\mathrm{C}}_{\mathrm{p}\mathrm{m}}$ and $\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{m}}}$ represent the contributions from moment-related power. After accounting for the combined influences of the wing and leading flap, the resulting forces, moments, and power generated by these components were summed.

The energy efficiency of the oscillating flat plate, denoted by η, is defined as the ratio of energy captured by the flapping foil to the energy of the incoming flow over one complete motion cycle. This can be mathematically expressed as follows [54]:

$$\eta ={\displaystyle \frac{\mathrm{P}}{\frac{1}{2}\rho {\mathrm{U}}_{\mathrm{\infty }}^3\mathrm{s} \mathrm{d}}}={\mathrm{C}}_{\mathrm{p}\mathrm{t}}{\displaystyle \frac{\mathrm{c}}{\mathrm{d}}}$$

(15)

where d represents the swept area of the flapping flat plate, which, in a two-dimensional context, corresponds to the vertical distance covered by the trailing edge of the flat plate. This study employed the overset mesh method due to its ability to simulate multibody structures and complex geometries [55]. The numerical simulations were performed using the commercial CFD software ANSYS Fluent 21 R1 [56]. The governing equations in this study are the unsteady, incompressible Reynolds-averaged Navier–Stokes equations. The finite volume methods, using pressure-based solvers, are detailed as follows:

$${\displaystyle \frac{\partial {\mathrm{u}}_{\mathrm{i}}}{\partial {\mathrm{x}}_{\mathrm{i}}}} = 0$$

(16)

$${\displaystyle \frac{\partial }{\partial \mathrm{t}}}(\rho {\mathrm{u}}_{\mathrm{i}}) + {\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}(\rho {\mathrm{u}}_{\mathrm{i}}{\mathrm{u}}_{\mathrm{j}}) = - {\displaystyle \frac{\partial \mathrm{P}}{\partial {\mathrm{x}}_{\mathrm{i}}}} + {\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}} (\mathsf{\mu }({\displaystyle \frac{\partial {\mathrm{u}}_{\mathrm{i}}}{\partial {\mathrm{x}}_{\mathrm{j}}}} + {\displaystyle \frac{\partial {\mathrm{u}}_{\mathrm{j}}}{\partial {\mathrm{x}}_{\mathrm{i}}}} - {\displaystyle \frac{2}{3}} {\mathsf{\delta }}_{\mathrm{i}\mathrm{j}} {\displaystyle \frac{\partial \mathrm{u}\mathrm{l}}{\partial \mathrm{x}\mathrm{l}}}) + {\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}} (-\rho \overline{{\mathrm{u}}_{\mathrm{i}}^{\prime }{\mathrm{u}}_{\mathrm{j}}^{\prime }}))$$

(17)

The Reynolds stress, denoted by −$\rho \overline{{\mathrm{u}}_{\mathrm{i}}^{\prime }{\mathrm{u}}_{\mathrm{j}}^{\prime }}$, is presented in Equation (17). This study used the k−ω SST turbulence model for the numerical simulations [55,57]. The dynamic mesh approach is implemented to model the heaving and pitching motions of the flapping flat plate. A pressure-velocity coupling algorithm is used to solve the equations. For discretization, both spatial and temporal terms are handled with second-order accuracy, and numerical computations are performed with double precision. The motion of the flat plate is defined through user-defined functions.

Figure 4 illustrates the computational domain used in this study, featuring a structured grid configuration covering an area of 70c by 50c for the numerical simulations. The pitching axis is positioned 25c away from the upstream boundary. A velocity inlet boundary condition is applied to the left side of the domain, while a pressure outlet boundary condition is set on the right. Slip boundary conditions are enforced on both lateral sides. Figure 4b shows the dynamic grids for both the wing and leading flap, and Figure 4c offers a closeup view of the primary wing structure and its leading flap. The grid configurations for the wing and leading flap were designed as oval grids with a diameter of 4c.

Grid- and time-independent studies were performed by adjusting the grid resolution and time step to evaluate precision and robustness under the specified boundary conditions. Mesh density was tested at three levels: coarse, medium, and fine.

Table 1. Examination of mesh resolution and time step independence.

Mesh Type	Moving Grid Elements	Background Elements	Time Step per Oscillation	${\mathbf{C}}_{\mathbf{p}\mathbf{t}}$	Mesh Variation ${\mathbf{C}}_{\mathbf{p}\mathbf{t}}$ (%)	Time Step Variation ${\mathbf{C}}_{\mathbf{p}\mathbf{t}}$ (%)	η (%)
Coarse	0.6 × ${10}^5$	0.3 × ${10}^5$	2000	0.891			34.53
Medium	1.2 × ${10}^5$	0.6 × ${10}^5$	500	0.904			35.03
			2000	0.887	0.44	1.88	34.37
			4000	0.883		0.45	34.22
Fine	2.6 × ${10}^5$	1.2 × ${10}^5$	2000	0.886	0.11		34.34

illustrates the effect of varying mesh resolutions and time intervals on the computed ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ values. The boundary conditions used in mesh- and time-independent studies were as follows: Reynolds number = 1100, reduced frequency = 0.14, and pitch angle = 75°. To assess time step independence, a medium-density mesh was used with time steps of 4000, 2000, and 500. Table 1 presents the power coefficients along with their associated variations. The results indicate that the medium-density mesh achieves sufficient accuracy, with only minor variations across different time steps.

This study validated the model by comparing its results with those presented in [30], demonstrating a high level of agreement, as shown in Figure 5. To optimize both accuracy and computational efficiency, a medium-sized grid with 2.0 × ${10}^3$ time steps was selected.

3. Findings and Analysis

The simulations were conducted under the following conditions: pivot point at $x_p$ = c/3, phase angle $\varnothing$ = 90°, reduced frequency $f^{\ast }$ = 0.14, and heaving amplitude ${\mathrm{h}}_{\mathrm{o}}$/c = 1.0, which has been shown to yield optimal power, as indicated in [54].

3.1. Effect of Hybrid and Sinusoidal Motions on the Wing Fitted with an Attached Leading Flap

The efficiency of the leading flap attached to the wing is evaluated in comparison to two setups as follows: the best case of the single flat plate using sinusoidal motion with a pitch angle of ${\theta }_{\mathrm{O}}$ = 75° [30,31] and the best case of a single flat plate using hybrid motion with a pitch angle of ${\theta }_{\mathrm{O}}$ set to 70° [51].

Table 2. Type of motions, pitch angles, and lengths of a wing with and without any attached flaps.

Setup Type	Flat Plate Without Flap [30,31]	Flat Plate Without Flap [51]	Wing With Leading Flap [32]
			Leading Flap	Wing
Motion type	Sinusoidal	Hybrid	Sinusoidal	Sinusoidal
Length percentage	100%	100%	40%	60%
Pitching angle	75°	70°	45°	95°
${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$	0.963	1.16	1.183

lists the pitch angles and lengths for these three cases. The maximum output power recorded for a single flat plate with sinusoidal motions is ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ = 0.963 [30,31]. For a wing with a leading flap using sinusoidal motion, the highest output power achieved is ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ = 1.183 [32]. In the case of a flat plate using hybrid motion, the peak output power is ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ = 1.16 as reported in [51].

This study presents the calculated results for three cases: (1) both the wing and leading flap employing hybrid motions (Case 1), (2) the leading flap using sinusoidal motion while the wing employs hybrid motion (Case 2), and (3) the leading flap using hybrid motion while the wing employs sinusoidal motion (Case 3). These cases are compared to Case 0, where both the leading flap and wing use sinusoidal motions. For all cases, the length percentages are set at 40% for the leading flap and 60% for the wing, with pitch angles of 45° for the leading flap and 95° for the wing. The findings presented in

Table 3. Lists the findings for an oscillating wing fitted with an attached leading flap at 40% and wing at 60% of c at ${\theta }_{\mathrm{O}}$= 95° and ${\psi }_{\mathrm{o}}$ = 45°.

Cases	Case 0	Case 1	Case 2	Case 3
Leading flap	Sinusoidal	Hybrid	Sinusoidal	Hybrid
Wing	Sinusoidal	Hybrid	Hybrid	Sinusoidal
${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$	1.544	1.451	1.403	1.488
${\mathrm{C}}_{\mathrm{p}\mathrm{m}}$	−0.306	−0.251	−0.252	−0.381
${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$	1.183	1.200	1.150	1.106
Δ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ %		1.43	−2.78	−6.50
η	43.01	40.24	38.56	40.18

reveal that the configuration in Case 1 outperforms both Case 2 and Case 3 in terms of performance. The calculated total power coefficients for Case 2 and Case 3 are even lower, at 1.15 and 1.106, respectively, compared to Case 0. Additionally, the efficiency in Cases 1, 2, and 3 is lower than in Case 0. Including only the top-performing cases allowed us to focus on the configurations that offer notable improvements.

For the calculations presented in

Table 4. Lists the power coefficients and efficiency for the wing with an attached leading flap using hybrid motions (case 1) at ${\theta }_{\mathrm{O}}$= 85° with different ${\psi }_{\mathrm{o}}$ values against two configurations.

Configuration Type	Flat Plate Without Flap Using Sinusoidal Motion [30,31]	Flat Plate Without Flap Using Hybrid Motion [51]	Case 1 (L35−W65)
Pitch angle	${\theta }_{\mathrm{O}}$	${\theta }_{\mathrm{O}}$	${\psi }_{\mathrm{o}}$	${\psi }_{\mathrm{o}}$	${\psi }_{\mathrm{o}}$
	75°	70°	40°	45°	50°
${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$	0.929	1.047	1.427	1.474	1.450
${\mathrm{C}}_{\mathrm{p}\mathrm{m}}$	0.034	0.114	−0.213	−0.198	−0.253
${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$	0.963	1.16	1.213	1.276	1.196
Δ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ %		20.45	25.96	32.50	24.19
η	37.30	40.53	41.71	43.87	41.12
Δη %		8.65	11.82	17.61	10.24

, the selected dimensions for the oscillating wing with an attached leading flap were as follows: the leading flap was set to 35% of the chord length, while the wing length was 65% of the chord length. In Case 1, where both the wing and leading flap use hybrid pitching motion, the wing’s pitch angle is set to ${\theta }_{\mathrm{O}}$ = 85°, while ${\psi }_{\mathrm{o}}$ is varied at 40°, 45°, and 50°. The calculated result at ${\psi }_{\mathrm{o}}$ = 45° achieves a ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ of 1.276 and an efficiency η of 43.87%, surpassing the performance of a single flat plate with sinusoidal pitching motion, as referenced in [30,31], and a single flat plate with hybrid pitching motion, as referenced in [51]. This results in a power gain of 32.50% compared to a single flat plate with sinusoidal motion, and a 10.0% increase compared to a single flat plate with hybrid motion. Additionally, the data reveal a 17.61% improvement in efficiency over a single flat plate with sinusoidal motion [31], and an 8.24% increase compared to a single flat plate with hybrid motion [51].

3.2. Effect of Various Pitching Motions on a Wing Featuring a Leading Flap

To investigate the effect of hybrid pitching motion, this study examined wing pitch angles of 80°, 85°, 90°, and 95°, with wing lengths ranging from 60% to 65% of the chord length. Additionally, the leading pitch angle ranged from 30° to 55°, with leading flap lengths from 35% to 40% of the chord.

Table 5. Type of motions, lengths, ${\theta }_0$, and ${\psi }_{\mathrm{o}}$.

Cases	Case 0		Case 1		Case 2
Configuration	Leading	Wing	Leading	Wing	Leading	Wing
Motion type	Sinusoidal	Sinusoidal	Hybrid	Hybrid	Sinusoidal	Hybrid
Length percentages %	40	60	35	65	40	60
Pitch angle	${\psi }_{\mathrm{o}}$ = 45°	${\theta }_0$ = 95°	${\psi }_{\mathrm{o}}$ = 45°	${\theta }_0$ = 85°	${\psi }_{\mathrm{o}}$ = 50°	${\theta }_0$ = 90°
${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$	1.183		1.276		1.276

presents the best cases of different pitching motion strategies: Case 1 utilizes hybrid pitching motion on both the leading flap and the wing, while Case 2 employs sinusoidal pitching motion on the leading flap and hybrid pitching motion on the wing. These cases are compared to Case 0, where sinusoidal pitching motion is used on both the leading flap and the wing.

In Case 0, which serves as the baseline configuration for the two bodies (a wing with a leading flap), the leading flap and wing have length percentages of 40% and 60%, respectively, with pitch angles of ${\psi }_{\mathrm{o}}$ = 45° for the leading flap and ${\theta }_{\mathrm{O}}$ = 95° for the wing, both using sinusoidal pitching motions. The best case of Case 1 has length percentages of 35% for the leading flap and 65% for the wing, with pitch angles of ${\psi }_{\mathrm{o}}$ = 45° for the leading flap and ${\theta }_{\mathrm{O}}$ = 85° for the wing, both using hybrid pitching motions. In the best case of Case 2, the leading flap and wing have length percentages of 40% and 60%, respectively, with a pitch angle of ${\psi }_{\mathrm{o}}$ = 50° for the leading flap and ${\theta }_{\mathrm{O}}$ = 90° for the wing, where sinusoidal pitching is applied to the leading flap and hybrid pitching to the wing. Both the best Cases, 1 and 2, achieved a similar total output power coefficient of ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ = 1.276, representing a 7.86% improvement over Case 0.

Table 6. The calculated results for an oscillating wing with a leading flap at different ${\psi }_{\mathrm{o}}$ values.

Cases	Case 0			Case 1			Case 2
Leading pitch angle ${\psi }_{\mathrm{o}}$	40°	45°	50°	40°	45°	50°	45°	50°	55°
Cpl	1.495	1.544	1.396	1.427	1.474	1.450	1.455	1.511	1.491
Cpm	−0.368	−0.360	−0.222	−0.213	−0.198	−0.253	−0.223	−0.234	−0.332
${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$	1.127	1.183	1.173	1.213	1.276	1.196	1.232	1.276	1.159
Δ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ %				2.53	7.86	1.09	4.14	7.86	−2.02
η	40.98	43.01	42.65	41.71	43.87	41.12	41.79	43.28	39.31
Δη					1.99			0.62

presents the best cases identified in this study. Both Case 1 and Case 2 demonstrate competitive output power across various configurations, highlighting their effectiveness in enhancing performance. With the pitch angle set to ${\psi }_{\mathrm{o}}$ = 45°, Case 0 shows a notable advantage in the pushing power coefficient (${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$) compared to Cases 1 and 2. The total power coefficient (${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$) reaches its highest value of 1.276 in Case 1 at a leading pitch angle of ${\psi }_{\mathrm{o}}$ = 45° and in Case 2 at ${\psi }_{\mathrm{o}}$ = 50°. The maximum efficiency is 43.87% in Case 1 and 43.28% in Case 2, both slightly higher than the 43.01% efficiency observed in Case 0.

Figure 6 illustrates the power coefficients and efficiencies for the best results of Case 1 and Case 2 compared to Case 0. Using hybrid motion for both the wing and leading flap (Case 1) or exclusively for the wing (Case 2) improves both output power and efficiency. Case 1 performed admirably, achieving an output power of 1.276 and an efficiency of 43.87%, resulting in a 7.86% increase in output power and a 1.99% improvement in efficiency. In comparison, Case 2 reached a similar output power of 1.276, reflecting the same 7.86% increase in output power, but had slightly lower efficiency at 43.28%, representing a 0.62% improvement. The calculated findings for Cases 1 and 2 are compared to the baseline case (Case 0), which utilizes sinusoidal pitching motion for both the leading flap and the wing. These results suggest that hybrid motion is advantageous, particularly when applied to both the wing and the leading flap.

3.3. Analyzing the Three Cases in More Detail

Figure 7 shows the findings for ${\mathrm{C}}_{\mathrm{L}}$, ${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$, ${\mathrm{C}}_{\mathrm{p}\mathrm{m}}$, and ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ under the conditions outlined in Table 5, showcasing the best results for various pitch motions and angles across Cases 0, 1, and 2.

Figure 7a shows that Case 2 (blue line) exhibits the strongest pushing force, ${\mathrm{C}}_{\mathrm{L}}$, with a noticeable difference from Case 0 (black line). In Case 1 (red line), the ${\mathrm{C}}_{\mathrm{L}}$ value is also higher than in Case 0, though it remains below the level observed in Case 2. Case 1 demonstrates a relatively smooth pattern in the pushing force coefficient, ${\mathrm{C}}_{\mathrm{L}}$, compared to the other two cases. Figure 7b shows that Cases 0, 1, and 2 exhibit similar ${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$ values. While Case 2 demonstrates a higher ${\mathrm{C}}_{\mathrm{L}}$ compared to Cases 0 and 1, the value of ${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$ is determined by the pushing force coefficient and heaving velocity. However, the similarity in ${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$ is attributed to the low heaving velocity at the initial time. Figure 7c shows that Cases 1 and 2, using hybrid pitching motion, exhibit smoother and more stable ${\mathrm{C}}_{\mathrm{p}\mathrm{m}}$ variations. In contrast, Case 0, with sinusoidal pitching, displays significant fluctuations. Figure 7d shows that Case 2 achieves the highest ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ at the initial time, indicating that hybrid pitching on both the wing and leading flap enhances total power output.

Figure 8 shows vorticity contour, while Figure 9 shows pressure distribution at various time steps, both under the same computational conditions outlined in Figure 7.

At t/T = 0.05, in Case 0, the vortex remains close to and attached to the upper surface, while in Case 2, it is farther away and detached from the upper surface at the hinge point of the flapping flat plate. At this moment, the pressure difference in Case 2 is greater than in Cases 0 and 1, with Case 1 also showing a higher pressure difference than Case 0 due to the use of hybrid motion in both Cases 1 and 2. Figure 9a indicates that Case 2 exhibits a greater pressure differential than Case 1, which can be attributed to its extended leading flap.

At t/T = 0.15, as shown in Figure 8b, the lower surface of the leading flap exhibits flow separation, with the extent varying by case. In Case 0, the separation occurs close to the hinge area and is smaller than in Cases 1 and 2. Meanwhile, the separation in Case 2 is notably larger than in Cases 0 and 1. The variation in the pressure coefficient along the lower surface, from x = −0.20 to the hinge point at x = 0.0, is shown in Figure 9b. In this region, pressure increases in Case 0, while it decreases in Case 2, with Case 1 displaying an intermediate behavior between Cases 0 and 2. These pressure changes are closely associated with the dynamics of flow separation and vortex size, as observed in Figure 8b at t/T = 0.15.

At t/T = 0.25, the flat plate reaches its maximum pitching angle: in Case 0, the wing is inclined at ${\theta }_0$ = 95°; in Case 1, the wing is inclined at ${\theta }_0$ = 85°; and in Case 2, the wing is positioned vertically at ${\theta }_0$ = 90°. This results in a shorter horizontal projection length in Case 0 compared to Cases 1 and 2, as shown in Figure 8 and Figure 9c at the same time.

As the flat plate descends between t/T = 0.25 and t/T = 0.5, it approaches a horizontal orientation. This motion, characterized by a gradual reduction in speed, contributes to the development of increased vorticity along the lower surface. At time 0.45, Figure 8 illustrates varying responses among the different cases, highlighting a counterclockwise vortex at the trailing edge. In Case 0, there is a noticeable detachment of vorticity along the lower surface. Figure 9d shows that in Cases 0 and 1, this vorticity formation leads to an increase in pressure on the lower surface near the rear edge.

4. Conclusions

Our studies were conducted through numerical analyses of the performance of an oscillating wing with an attached leading flap harvester that employed hybrid motions. These transient simulations used the overset mesh method along with the k−ω SST turbulence model. The leading flap lengths varied between 30% and 40% of the chord, with pitch angles ranging from 30° to 60°. The wing was analyzed with lengths ranging from 60% to 70% of the chord and pitch angles ranging between 80° and 95°. This study showed that optimal output power and efficiency were achieved when both the wing and leading flap employed hybrid pitching motions. The most effective configuration involved a leading flap length of 35% of the chord, with a pitch angle of 45° and a wing length of 65% with a pitch angle of 85°. This setup led to a notable improvement in power output, increasing it by 32.50% compared to a single flat plate executing sinusoidal motion. Using hybrid motions for both the wing and leading flap resulted in a maximum efficiency of 43.87%. This study underscores the advantages of implementing hybrid motions.